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How to break SO(10) via SO(4) × SO(6) down to SU(2)L × SU(3)c × U(1)
9 July 1981
PHYSICS LETTERS
Volume 103B, number 1
HOW TO BREAK SO(10) VIA SO(4) X SO(6) DOWN TO SU(2), Masaki YASUE
X SU(3)C X U(1)
’
Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan
Received 13 October 1980 Revised manuscript received 9 February 1981
A clear discussion is presented in order to see the spontaneous symmetry breaking of SO(10) down to Sum X SU(3), U(1) via SO(4) X SO(6), the pattern of which is determined by the Higgs potential. Special attention is paid to the stability of the vacuum specified by an adjoint (45) representation. The Higgs system should at least consist of a symmetric {54}, a spinorial (16) and the (45). X
The unified Majorana (L-R)
model
neutrino
symmetry.
[l]
has various
value of sin2ew
These symmetries
How does SO(l0) Symmetry
based on SO(10)
[2], a reasonable
attractive
aspects.
[3] and possesses
are due to the appearance
It predicts
the existence
a quark-lepton
of the subgroup
symmetry
of the light and a left-right
SO(4) X SO(6) of SO(10)
*l.
break down via SO(4) X SO(6)?
breaking
is expected
to be brought
out by a set of Higgs scalars each of which
develops
a non-vanish-
ing vacuum expectation value (vev). The original symmetry of SO(10) can be spontaneously broken down to SO(4) X SO(6) by a symmetric {54} tensor and/or an antisymmetric {210} tensor since these contain the SO(4) X SO(6) singlets: {54} = (1, 1; I)+
(3,3,
under
X SU(2),
the SU(2),
l)+
(2, 2;6)
+(l,
1;20),
X SU(4) decomposition.
An adjoint
{45} representation
whose
decomposition
is
{45}=(3,1;1)+(1,3;1)+(2,2;6)+(1,1;15), can break SO(10) down to SO(4) X SU(3), X U(l)B_L by the (1, 1; 15) and SU(2), X U(l)R (1, 3; 1). To break SU(2)R completely, we must employ a spinorial { 16) representation: (16)
= (2, 1;4)
Collecting metry
+ (1, 2,4*).
these scalars, one can find that SO(10)
breaking
X SU(4) by the
at the final stage is furnished
spontaneously
breaks down to SU(2),
by a Higgs scalar which
SU(2), X SU(2), X SU(3), X U(l)&L. Th ere f ore, the candidates symmetric odd rank tensors, { 1ZO} and { 126).
contains
X U(1) X SU(3),.
(2, 1; 1”) and/or
are the { 16}, a vectorial
Sym-
(2, 2; 1 Cbunder
{ 10) and other anti-
The symmetry breaking pattern which actually occurs should be determined by finding residual symmetries sitting on the absolute minimum of a given Higgs potential. General discussions of performing this program have been presented in ref. [5] and will be in ref. [6] by taking the Higgs system with the 145) and (16) and with the 1 Fellow of the Japan Society for the Promotion of Science. *’ These imply some kind of connection between SO(10) and the composite structure of quarks and leptons 141. 0 031-9163/81/0000-0000/$02.50
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9 July 1981
PHYSICS LETTERS
Volume 103B, number 1
(45) and {54}, respectively. In this note, we explain the requisite Higgs system bringing out the spontaneous symmetry breaking of SO(10) down to SU(2), X SU(3)e X U(1) via SO(4) X SO(6) m a simple way and demonstrate that the minimal set of the Higgs scalars to generate this symmetry breaking pattern consists of the {54}, (4.5) and { 16). Let us concentrate ourselves on discussing how to break SO(10) down to SU(2), X SU(3)c X U(1) accompanied by SO(4) X SO(6), SO(4) and/or SO(6) in the course of symmetry breaking, based on those analyses
[5,61. For later convenience, the mass spectra of the Higgs system with the (45) alone are exhibited: SU(2), X U(l)R X SU(4): 4h&
for
(3; 1)
; -2X20;
where the vev, tiR, transforms x S”(3), -2x&
for as (1,3;
in the case of
(1; 15)
(la)
1) under SU(2),
X SU(2),
X SU(4); and in the case of SU(2),
X SU(2)R
x u( l)B _ ,_ : for
(3, 1; 1’)
;
4X2wc
for
(1, 1; 8’)
(lb)
where the vev, wy , transforms as (1, 1; 15). The quartic coupling h, (> 0) is to be defined by (2). The two vev’s wR and oy are chosen in such a way that SU(5) X U(1) is recovered if WR = wy . The negative mass is a signal of the instability of the vacuum. It must be made to be positive through coupling to other Higgs scalars which are taken to be the { 16) and the {54}. First, the Higgs system with the (45) and { 16) is examined. A relevant symmetry breaking pattern is expected if the (45) (@,J first breaks SO(10) down to SO(4) X SU(3), X U(l)B_L or SU(2), X U(l)R X SU(4), which is in turn broken down to SU(2), X SU(3), X U(1) by the { 16) (x). The Higgs potential is given by [5] V= -+j~*
11~11 +$hI
11~112 +:X2
trQ4) -$v*
(x’Fx)+$A3
(x1x)*
l~~4(X^I,,X)~~Y~X)+~~X+X~II~ll~P~X+~2X~: (2)
where
The conditions h,>O,
to allow SO(10) to break down to SU(2),
x,>o,
X SU(3), X U(1) are
p>o.
(3)
The other constraint can be obtained by imposing the positivity condition on the physical Higgs-scalar masses within the tree approximation of the potential. The masses for the Higgs transforming as (1, SC) and (3, 1”) under SU(2), X SU(3), X U(1) are found to be 2h,(w,
+ 2wy)(wy
- WI&
for (1, 8’);
2h,(w,
+ 2wR)(wR
- wy),
for
(3, 1”).
(4a, b)
These two expressions can be interchanged by WR ++wy . It is reasonable because aR and wy are nothing but labelling the vev’s as they are and both preserve SU(2): SU(2), and SU(2) of SU(3)e; therefore, (4a) and (4b) are symmetric under interchange in this sense. One can readily understand that one expression of (4a) and (4b) turns out to be negative if either &+ % wy respecting SO(6) or wy 3 c+ respecting SO(4) is imposed. The fact that WY - uR reflects the W(5) singlet structure of the vev of the {I6}. Thus, it cannot have SO(4) or SO(6) in the Higgs system with the {45} and { 16). Instead, it has a hierarchy SO(10) + SU(5) + SU(2), X SU(3)e X U(1) [5] . Next, let us consider the Higgs system with the (45) (GPLy)and (54) (Q,,,). The Higgs potential is given by [6] V= -in*
II@11+$A1
lIdI ++A2 tr(G4)-fv*
ll~,ll ++A3 11~,112 +$A4 W,4) (5)
34
9 July 1981
PHYSICS LETTERS
Volume 103B. number 1
and (@,I,,= 4spv= Gsvp P,V
with trace condition tr(@,) = 0 *2 . In order to break SO(10) down to SO(4) X SU(3)e X U(l)B_L, SU(2), X U(l)R X SO(6) or SU(2), X U(l)R X SU(3)c X U(l)B _L, the following conditions must be fulfilled:
X,>O,
h,>O,
(6)
P,>O,
and hzwi
G(P,
+ P,)c$,
x&
G -%P,
(7a, b)
+ P&Y
or $(W;
-w$)=$(O1
(7c)
+P&.
The vev’s, tiR, wY and ws, are arranged by (@J = diag(3wsI/5,
3wJ/S,
-20sI/S,
-2wJ/S,
-2wsI/.5)
(8a)
and (@)= diag(w,A,
o,A,
+A,
w,A,
w~A),
@b)
where I=(;
;)>
A=(_;
;)-
(9)
The last condition of (6) (& > 0) ensures that the off-block diagonal elements of the {45} vanish at the minimum of the potential *3 after the (54) is transformed to the diagonal form (8a) by the SO(10) rotation. The constraints (7a), (7b) and (7~) are derived by minimizing the potential with respect to the {45}. The sign of fll + pZ can be determined from (6) and (7a, b, c): p, +/?:!>O
if
wg>o$(=O);
p1+/32<0
if
G$>tii(=O).
(1 Oa, b)
The masses of the physical Higgs scalars are calculated to be: 4XZwi
for (3; 1);
in the case of sU(2)L
-2h2w$
x U(1)R
- 4@, +p,)wi/5
in the case of SU(2), 4x&
-2hZwi
for (3, 1’);
+ 4@, + fl,)c$/S
for (1; 1.5),
(1 la, b)
x sU(4),
for (3, 1; 1’) and (1,3;
I’);
4AZw$
for (I, 1; 8’)
(12a, b)
X SU(2)R X SU(3)c X U(l), and 4X2w$
for (1, SC),
(13a, b)
in the case of SU(2), X U(l)R X SU(3), X U(~)&L. The first two cases must be compared with that with the {45} alone. Due to the constraints (7a) and (7b), the negative masses in (la) and (1 b) can be made to be positive. In the third case, positivity is dynamically engured by the relation between wy , uR and ws given by (7~).
*’ The inclusion
of the cubic term Astr(@i) seems to be inevitable for the case of (7a) and (7b) in order to obtain the absolute minimum respecting the physically interesting symmetries. However, in the case of (7c), we do not necessarily include the cubic term, (see appendix). Note that the local minimum can be achieved in the absence of the cubic coupling. Also allowed is a cubic term such as tr(@,e2). It is not relevant to the subject of the present discussion. *3 As long as the off-block diagonal elements are treated as a small perturbation to the system described by (Ba) and (8b).
3.5
9 July 1981
PHYSICS LETTERS
Volume 103B, number 1
The gauge hierarchies brought by the (45) and {54} are precisely what we are expecting. The constraints b, c) allow us to meet the hierarchies characterized by wz s wi, wi B w$, w: - wi 3 w$ and wf - w$ % c&. These are described by so(4)
’ s”(3)e
(7a,
’ ‘(l)B _L
SO(10) + SO(4) X SO(6)
(14)
E
SU(2),
X U(l)R X SO(6)
satisfying (7a) and (7b) and
s0(4) x s”(3)c ’ SO(10) -
SU(2), SU(2),
X U(l)R
x u(l)R
’ s”(3)c
x”(l)B_t,’
(15)
X SO(6)
satisfying (7c) *4 . The SO(6) symmetry shows up if PI + f12 > 0 (lOa) and the SO(4) symmetry < 0 (lob). Further condition arises through looking into the mass parameter ~2 given by P2 = (hl + h,t) II@11 - 2[a! + @I + P,)sl lldg
does if PI + fiz
(16)
,
where t = l/4 for (7a), l/6 for (7b) and 1/lO for (7~) and s = 3/20 for (7a), 1/15 for (7b) and 1/lO for (7~) (appendix). The (4.5) spontaneously breaks SO(10) if h, + X,t > 0.
(17)
The requisite condition a+ Co1 +P2s)<
0
to obtain ]]@,]1B I]@]](14) is found to be
if p2
Cl& b)
- W~,ll).
The remaining case (15) is only subject to the usual condition ~2 > 0 because I]$11- 11Gsl]. The breaking of the last residual symmetries in (14) and (15) can be furnished by the { 16) on a much smaller mass scale. A contribution from the { 16) to the masses of the physical Higgs scalars which welare discussing is minor compared with those from the (45) and (54) since the { 16) does not contain the (3, lc) and (1, 8c) in itself. A significant effect of the inclusion of the { 16} goes into the masses of the scalars transforming as 10 of SU(5)
[51. Summarizing our discussion, we have shown that the Higgs system with the {45} and { 16) does not seem to possess SO(4) or SO(6) in the course of the symmetry breaking of SO(10). To break SO(10) spontaneously down to SU(2), X SU(3), X U(1) via SO(4) and/or SO(6), the Higgs system at least includes the { 54} as well as the (45) and { 16). As shown in ref. [5], breaking SU(2), by means of the vev of the { 16) does not seem to correspond to the absolute minimum of the relevant Higgs potential and even to the local minimum unless there is a CUbit coupling of { 16*} { 16) {45}; therefore, we need the {lo} or else to break SU(2), completely. We then conclude that the minimal set of the Higgs scalars consists of the {54}, {45}, { 16) and { 10). The {54} could be replaced by an antisymmetric 4th rank tensor { 210} which also contains the SO(4) X SO(6) singlet component and similarly, one could employ an antisymmetric 5th rank tensor { 126) instead of the { 16) because a { 126) contains the SU(5) singlet. The symmetry breaking pattern of (14) is found to occur when the Higgs potential reaches its absolute minimum under the relevant condition (6) and (lOa, b) [6] . The three mass scales MG, MR and MC which appear in the symmetry breaking of SO(10) described by s”(4) SO(4) X SO(6) - Mc c MG SU(2),
’ s”(3)C ’ ‘(l)B
MR
_L z -
SO(10) +
X U(l)R
X SO(6) _I
SU(2),
X SU(3)e X U(1)
(19)
MC
*4 It is not clear in what circumstances this pattern corresponds to the absolute minimum of the potential. Only assured is that of SO(10) directly broken down to SU(2)L X U(l)R X SU(3), X U(l)B_L [6].
36
Volume
103B, number
can be bounded.
PHYSICS
1
9 July 1981
LETTERS
The detailed analysis is found in refs. [7,3].
The author thanks Professor J. Arafune for several helpful discussions and Professor H. Terazawa for careful reading of the manuscript. Suggestive discussions with all other members of the theory division at INS are also acknowledged. Appendix. The series of symmetry breaking patterns including the pattern via SO(4) X SO(6) are those via SO(2) X SO(8), SO(4) X SO(6) and SO(5) X SO(5) [8] *‘. The conditions (7a-c) are-replaced by the general ones: (i) fi ]1$,]/ > X(10 - 21) if tiR # 0 and wy = 0, (ii) p []@,I1
- WC)@ ~ Z)/(S - 21)
where - 511lIdI.
X(x) = @,/4)]x/(x
and(wR, WR, oy, oy, wy) for Z= 2 with respect to the matrix and diag(q9 = (WR, wy , wy,wy,oy)forZ=l A given by (9). No restriction is for the case via SO(5) X SO(5) because both {45}and (541, respectively, develop one vev. The function F(n) which does not depend on only the norms ]]@I]and 11$,11in the Higgs potential (5) is given by F(n) =f$
F&4 =$A4 m
~(n)llc?a2 - P(n)s(n>11~11 II@,11 +q4
llq12 +&
c’3’(n>119,113’2,
where n = 2,4 and 5 and @z) = CC4)(n) - {8[@)]
2/5X2~4} A(n),
C(3)(n) = [(lo - n)2 - n2] /[1000n(10 A(n) = (5 ~ n)2/n(10 Z-Z
- n)
0
-n)]
Ct4)(n) = [n3 + (10 - n)3] /lOOn(lO
-n),
U2 )
for (iii) for the others
and s(n) = (10 - n)/lO n
for (i)
= 12/[10(10 - n)]
for (ii),
= l/IO
for the others,
t(n) = l/n
for (i)
= l/(10 - n)
for (ii),
= l/10
for the others,
*’ Other symmetry breaking
patterns such as those via SO(9), SO(7) X SO(3) requires a set of equations satisfied by the parameters, which means that one should a priori prepare these coupling parameters in such a way that they satisfy the equation derived after minimizing the Higgs potential.
37
Volume
103B, number
/3(n) = PI + /3, =p, -b,
PHYSICS
1
for n = 2,4, forn=S.
For the case of (i) and (ii) (n = 2 and 4), the function F=$h,
9 July 1981
LETTERS
F becomes
~(n)ll~l12 - P(n)s(n) 11~11 ll~,ll +FI(n).
Because F,(n) involves the cubic term h, tr(@), F1 (n ) can reach its minimum at n = 4 for a certain range of the parameters h, and h, [6] . Therefore, the absolute minimum of F can occur at n = 4 although it does depend on 1]@]]/[]@,]1. The situation gets more complicated if the cubic term is absent because Fl(n) always takes its minimum at n = 5 [due to the term Z(4)(n)] . Therefore, if ]I$,113 ]]q5]],then Fl(n) governs the behavior of F, which allows the absolute minimum of the potential at n = 4. On the other hand, for the case of (iii) (n = 2 and 4) the function F is given by
F= ~h2114211 - &,LV>IIdI II@,11 +FI@). The relevant symmetry breaking pattern via SO(4) X SO(6) (n = 4) IS realized even in the absence of the cubic term h, tr(@) because A(5) < A(4) < A(2) and then the term proportional to A(n) plays the same role as that of the cubic term Cc3)(n) [because h, > 0, h, > 0; see (6)] . The explicit conditions are 534) < F(2), Z(5)
andDl+P2>B1
-P,>O
or&
-P2
leading to p2 > 0 *6 . The inclusion of the cubic term makes the vacuum more stable if X4 and h, are appropriately adjusted so as to produce the absolute minimum at n = 4. Finally, notice that for the breaking pattern via SO(5) X SO(5), the vev of @is given by diag (@ = (u, -u, U, -u, U, -u, U, -u, U, -u) in the basis of (Sa) and (8b). *6 The condition of p2 > 0 stated in footnote 3 is actually for the case (i) and (ii). This case (iii) automatically ment in footnote 3 because h2 > 0 if we assume II@11 > It@sll, which is related to footnote 4.
References [l]
[2]
[3] [4] [5] [6] [7] [8]
38
H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193; M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, Nucl. Phys. B155 (1979) 52. P. Ramond, Caltech preprint CALT-68-709 (February 1979); R. Barbieri et al., Phys. Lett. YOB (19 80) 9 1; E. Witten, Phys. Lett. YlB (1980) 81; M. Magg and Ch. Wetterich, Phys. Lett. 94B (1980) 61. Q. Shafi and Ch. Wetterich, Phys. Lett. 85B (1979) 52; Q. Shafi, M. Sondermann and Ch. Wetterich, Phys. Lett. 92B (1980). M. Yasub, Phys. Lett. YlB (1980) 85. F. Buccella, H. Ruegg and CA. Savoy, Phys. Lett. 94B (1980) 49 1; M. Yasue, INS-preprint INS-rep. -387 (August 1980); to be published in Phys. Rev. D (1981). M. Yasue, in preparation; G. Lazarides, M. Magg and Q. Shafi, Phys. Lett. 97B (1980) 87. M. Yasue, Prog. Theor. Phys. 65 (1981) 708; G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181 287. G. Lazarides, M. Magg and Q. Shafi, Phys. Lett. 97B (1980) 87; F. Buccella, H. Ruegg and C.A. Savoy, Nucl. Phys. B169 (1980) 68.
ensures
the state-
PHYSICS LETTERS
Volume 103B, number 1
HOW TO BREAK SO(10) VIA SO(4) X SO(6) DOWN TO SU(2), Masaki YASUE
X SU(3)C X U(1)
’
Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188, Japan
Received 13 October 1980 Revised manuscript received 9 February 1981
A clear discussion is presented in order to see the spontaneous symmetry breaking of SO(10) down to Sum X SU(3), U(1) via SO(4) X SO(6), the pattern of which is determined by the Higgs potential. Special attention is paid to the stability of the vacuum specified by an adjoint (45) representation. The Higgs system should at least consist of a symmetric {54}, a spinorial (16) and the (45). X
The unified Majorana (L-R)
model
neutrino
symmetry.
[l]
has various
value of sin2ew
These symmetries
How does SO(l0) Symmetry
based on SO(10)
[2], a reasonable
attractive
aspects.
[3] and possesses
are due to the appearance
It predicts
the existence
a quark-lepton
of the subgroup
symmetry
of the light and a left-right
SO(4) X SO(6) of SO(10)
*l.
break down via SO(4) X SO(6)?
breaking
is expected
to be brought
out by a set of Higgs scalars each of which
develops
a non-vanish-
ing vacuum expectation value (vev). The original symmetry of SO(10) can be spontaneously broken down to SO(4) X SO(6) by a symmetric {54} tensor and/or an antisymmetric {210} tensor since these contain the SO(4) X SO(6) singlets: {54} = (1, 1; I)+
(3,3,
under
X SU(2),
the SU(2),
l)+
(2, 2;6)
+(l,
1;20),
X SU(4) decomposition.
An adjoint
{45} representation
whose
decomposition
is
{45}=(3,1;1)+(1,3;1)+(2,2;6)+(1,1;15), can break SO(10) down to SO(4) X SU(3), X U(l)B_L by the (1, 1; 15) and SU(2), X U(l)R (1, 3; 1). To break SU(2)R completely, we must employ a spinorial { 16) representation: (16)
= (2, 1;4)
Collecting metry
+ (1, 2,4*).
these scalars, one can find that SO(10)
breaking
X SU(4) by the
at the final stage is furnished
spontaneously
breaks down to SU(2),
by a Higgs scalar which
SU(2), X SU(2), X SU(3), X U(l)&L. Th ere f ore, the candidates symmetric odd rank tensors, { 1ZO} and { 126).
contains
X U(1) X SU(3),.
(2, 1; 1”) and/or
are the { 16}, a vectorial
Sym-
(2, 2; 1 Cbunder
{ 10) and other anti-
The symmetry breaking pattern which actually occurs should be determined by finding residual symmetries sitting on the absolute minimum of a given Higgs potential. General discussions of performing this program have been presented in ref. [5] and will be in ref. [6] by taking the Higgs system with the 145) and (16) and with the 1 Fellow of the Japan Society for the Promotion of Science. *’ These imply some kind of connection between SO(10) and the composite structure of quarks and leptons 141. 0 031-9163/81/0000-0000/$02.50
0 North-Holland
Publishing
Company
33
9 July 1981
PHYSICS LETTERS
Volume 103B, number 1
(45) and {54}, respectively. In this note, we explain the requisite Higgs system bringing out the spontaneous symmetry breaking of SO(10) down to SU(2), X SU(3)e X U(1) via SO(4) X SO(6) m a simple way and demonstrate that the minimal set of the Higgs scalars to generate this symmetry breaking pattern consists of the {54}, (4.5) and { 16). Let us concentrate ourselves on discussing how to break SO(10) down to SU(2), X SU(3)c X U(1) accompanied by SO(4) X SO(6), SO(4) and/or SO(6) in the course of symmetry breaking, based on those analyses
[5,61. For later convenience, the mass spectra of the Higgs system with the (45) alone are exhibited: SU(2), X U(l)R X SU(4): 4h&
for
(3; 1)
; -2X20;
where the vev, tiR, transforms x S”(3), -2x&
for as (1,3;
in the case of
(1; 15)
(la)
1) under SU(2),
X SU(2),
X SU(4); and in the case of SU(2),
X SU(2)R
x u( l)B _ ,_ : for
(3, 1; 1’)
;
4X2wc
for
(1, 1; 8’)
(lb)
where the vev, wy , transforms as (1, 1; 15). The quartic coupling h, (> 0) is to be defined by (2). The two vev’s wR and oy are chosen in such a way that SU(5) X U(1) is recovered if WR = wy . The negative mass is a signal of the instability of the vacuum. It must be made to be positive through coupling to other Higgs scalars which are taken to be the { 16) and the {54}. First, the Higgs system with the (45) and { 16) is examined. A relevant symmetry breaking pattern is expected if the (45) (@,J first breaks SO(10) down to SO(4) X SU(3), X U(l)B_L or SU(2), X U(l)R X SU(4), which is in turn broken down to SU(2), X SU(3), X U(1) by the { 16) (x). The Higgs potential is given by [5] V= -+j~*
11~11 +$hI
11~112 +:X2
trQ4) -$v*
(x’Fx)+$A3
(x1x)*
l~~4(X^I,,X)~~Y~X)+~~X+X~II~ll~P~X+~2X~: (2)
where
The conditions h,>O,
to allow SO(10) to break down to SU(2),
x,>o,
X SU(3), X U(1) are
p>o.
(3)
The other constraint can be obtained by imposing the positivity condition on the physical Higgs-scalar masses within the tree approximation of the potential. The masses for the Higgs transforming as (1, SC) and (3, 1”) under SU(2), X SU(3), X U(1) are found to be 2h,(w,
+ 2wy)(wy
- WI&
for (1, 8’);
2h,(w,
+ 2wR)(wR
- wy),
for
(3, 1”).
(4a, b)
These two expressions can be interchanged by WR ++wy . It is reasonable because aR and wy are nothing but labelling the vev’s as they are and both preserve SU(2): SU(2), and SU(2) of SU(3)e; therefore, (4a) and (4b) are symmetric under interchange in this sense. One can readily understand that one expression of (4a) and (4b) turns out to be negative if either &+ % wy respecting SO(6) or wy 3 c+ respecting SO(4) is imposed. The fact that WY - uR reflects the W(5) singlet structure of the vev of the {I6}. Thus, it cannot have SO(4) or SO(6) in the Higgs system with the {45} and { 16). Instead, it has a hierarchy SO(10) + SU(5) + SU(2), X SU(3)e X U(1) [5] . Next, let us consider the Higgs system with the (45) (GPLy)and (54) (Q,,,). The Higgs potential is given by [6] V= -in*
II@11+$A1
lIdI ++A2 tr(G4)-fv*
ll~,ll ++A3 11~,112 +$A4 W,4) (5)
34
9 July 1981
PHYSICS LETTERS
Volume 103B. number 1
and (@,I,,= 4spv= Gsvp P,V
with trace condition tr(@,) = 0 *2 . In order to break SO(10) down to SO(4) X SU(3)e X U(l)B_L, SU(2), X U(l)R X SO(6) or SU(2), X U(l)R X SU(3)c X U(l)B _L, the following conditions must be fulfilled:
X,>O,
h,>O,
(6)
P,>O,
and hzwi
G(P,
+ P,)c$,
x&
G -%P,
(7a, b)
+ P&Y
or $(W;
-w$)=$(O1
(7c)
+P&.
The vev’s, tiR, wY and ws, are arranged by (@J = diag(3wsI/5,
3wJ/S,
-20sI/S,
-2wJ/S,
-2wsI/.5)
(8a)
and (@)= diag(w,A,
o,A,
+A,
w,A,
w~A),
@b)
where I=(;
;)>
A=(_;
;)-
(9)
The last condition of (6) (& > 0) ensures that the off-block diagonal elements of the {45} vanish at the minimum of the potential *3 after the (54) is transformed to the diagonal form (8a) by the SO(10) rotation. The constraints (7a), (7b) and (7~) are derived by minimizing the potential with respect to the {45}. The sign of fll + pZ can be determined from (6) and (7a, b, c): p, +/?:!>O
if
wg>o$(=O);
p1+/32<0
if
G$>tii(=O).
(1 Oa, b)
The masses of the physical Higgs scalars are calculated to be: 4XZwi
for (3; 1);
in the case of sU(2)L
-2h2w$
x U(1)R
- 4@, +p,)wi/5
in the case of SU(2), 4x&
-2hZwi
for (3, 1’);
+ 4@, + fl,)c$/S
for (1; 1.5),
(1 la, b)
x sU(4),
for (3, 1; 1’) and (1,3;
I’);
4AZw$
for (I, 1; 8’)
(12a, b)
X SU(2)R X SU(3)c X U(l), and 4X2w$
for (1, SC),
(13a, b)
in the case of SU(2), X U(l)R X SU(3), X U(~)&L. The first two cases must be compared with that with the {45} alone. Due to the constraints (7a) and (7b), the negative masses in (la) and (1 b) can be made to be positive. In the third case, positivity is dynamically engured by the relation between wy , uR and ws given by (7~).
*’ The inclusion
of the cubic term Astr(@i) seems to be inevitable for the case of (7a) and (7b) in order to obtain the absolute minimum respecting the physically interesting symmetries. However, in the case of (7c), we do not necessarily include the cubic term, (see appendix). Note that the local minimum can be achieved in the absence of the cubic coupling. Also allowed is a cubic term such as tr(@,e2). It is not relevant to the subject of the present discussion. *3 As long as the off-block diagonal elements are treated as a small perturbation to the system described by (Ba) and (8b).
3.5
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PHYSICS LETTERS
Volume 103B, number 1
The gauge hierarchies brought by the (45) and {54} are precisely what we are expecting. The constraints b, c) allow us to meet the hierarchies characterized by wz s wi, wi B w$, w: - wi 3 w$ and wf - w$ % c&. These are described by so(4)
’ s”(3)e
(7a,
’ ‘(l)B _L
SO(10) + SO(4) X SO(6)
(14)
E
SU(2),
X U(l)R X SO(6)
satisfying (7a) and (7b) and
s0(4) x s”(3)c ’ SO(10) -
SU(2), SU(2),
X U(l)R
x u(l)R
’ s”(3)c
x”(l)B_t,’
(15)
X SO(6)
satisfying (7c) *4 . The SO(6) symmetry shows up if PI + f12 > 0 (lOa) and the SO(4) symmetry < 0 (lob). Further condition arises through looking into the mass parameter ~2 given by P2 = (hl + h,t) II@11 - 2[a! + @I + P,)sl lldg
does if PI + fiz
(16)
,
where t = l/4 for (7a), l/6 for (7b) and 1/lO for (7~) and s = 3/20 for (7a), 1/15 for (7b) and 1/lO for (7~) (appendix). The (4.5) spontaneously breaks SO(10) if h, + X,t > 0.
(17)
The requisite condition a+ Co1 +P2s)<
0
to obtain ]]@,]1B I]@]](14) is found to be
if p2
Cl& b)
- W~,ll).
The remaining case (15) is only subject to the usual condition ~2 > 0 because I]$11- 11Gsl]. The breaking of the last residual symmetries in (14) and (15) can be furnished by the { 16) on a much smaller mass scale. A contribution from the { 16) to the masses of the physical Higgs scalars which welare discussing is minor compared with those from the (45) and (54) since the { 16) does not contain the (3, lc) and (1, 8c) in itself. A significant effect of the inclusion of the { 16} goes into the masses of the scalars transforming as 10 of SU(5)
[51. Summarizing our discussion, we have shown that the Higgs system with the {45} and { 16) does not seem to possess SO(4) or SO(6) in the course of the symmetry breaking of SO(10). To break SO(10) spontaneously down to SU(2), X SU(3), X U(1) via SO(4) and/or SO(6), the Higgs system at least includes the { 54} as well as the (45) and { 16). As shown in ref. [5], breaking SU(2), by means of the vev of the { 16) does not seem to correspond to the absolute minimum of the relevant Higgs potential and even to the local minimum unless there is a CUbit coupling of { 16*} { 16) {45}; therefore, we need the {lo} or else to break SU(2), completely. We then conclude that the minimal set of the Higgs scalars consists of the {54}, {45}, { 16) and { 10). The {54} could be replaced by an antisymmetric 4th rank tensor { 210} which also contains the SO(4) X SO(6) singlet component and similarly, one could employ an antisymmetric 5th rank tensor { 126) instead of the { 16) because a { 126) contains the SU(5) singlet. The symmetry breaking pattern of (14) is found to occur when the Higgs potential reaches its absolute minimum under the relevant condition (6) and (lOa, b) [6] . The three mass scales MG, MR and MC which appear in the symmetry breaking of SO(10) described by s”(4) SO(4) X SO(6) - Mc c MG SU(2),
’ s”(3)C ’ ‘(l)B
MR
_L z -
SO(10) +
X U(l)R
X SO(6) _I
SU(2),
X SU(3)e X U(1)
(19)
MC
*4 It is not clear in what circumstances this pattern corresponds to the absolute minimum of the potential. Only assured is that of SO(10) directly broken down to SU(2)L X U(l)R X SU(3), X U(l)B_L [6].
36
Volume
103B, number
can be bounded.
PHYSICS
1
9 July 1981
LETTERS
The detailed analysis is found in refs. [7,3].
The author thanks Professor J. Arafune for several helpful discussions and Professor H. Terazawa for careful reading of the manuscript. Suggestive discussions with all other members of the theory division at INS are also acknowledged. Appendix. The series of symmetry breaking patterns including the pattern via SO(4) X SO(6) are those via SO(2) X SO(8), SO(4) X SO(6) and SO(5) X SO(5) [8] *‘. The conditions (7a-c) are-replaced by the general ones: (i) fi ]1$,]/ > X(10 - 21) if tiR # 0 and wy = 0, (ii) p []@,I1
- WC)@ ~ Z)/(S - 21)
where - 511lIdI.
X(x) = @,/4)]x/(x
and(wR, WR, oy, oy, wy) for Z= 2 with respect to the matrix and diag(q9 = (WR, wy , wy,wy,oy)forZ=l A given by (9). No restriction is for the case via SO(5) X SO(5) because both {45}and (541, respectively, develop one vev. The function F(n) which does not depend on only the norms ]]@I]and 11$,11in the Higgs potential (5) is given by F(n) =f$
F&4 =$A4 m
~(n)llc?a2 - P(n)s(n>11~11 II@,11 +q4
llq12 +&
c’3’(n>119,113’2,
where n = 2,4 and 5 and @z) = CC4)(n) - {8[@)]
2/5X2~4} A(n),
C(3)(n) = [(lo - n)2 - n2] /[1000n(10 A(n) = (5 ~ n)2/n(10 Z-Z
- n)
0
-n)]
Ct4)(n) = [n3 + (10 - n)3] /lOOn(lO
-n),
U2 )
for (iii) for the others
and s(n) = (10 - n)/lO n
for (i)
= 12/[10(10 - n)]
for (ii),
= l/IO
for the others,
t(n) = l/n
for (i)
= l/(10 - n)
for (ii),
= l/10
for the others,
*’ Other symmetry breaking
patterns such as those via SO(9), SO(7) X SO(3) requires a set of equations satisfied by the parameters, which means that one should a priori prepare these coupling parameters in such a way that they satisfy the equation derived after minimizing the Higgs potential.
37
Volume
103B, number
/3(n) = PI + /3, =p, -b,
PHYSICS
1
for n = 2,4, forn=S.
For the case of (i) and (ii) (n = 2 and 4), the function F=$h,
9 July 1981
LETTERS
F becomes
~(n)ll~l12 - P(n)s(n) 11~11 ll~,ll +FI(n).
Because F,(n) involves the cubic term h, tr(@), F1 (n ) can reach its minimum at n = 4 for a certain range of the parameters h, and h, [6] . Therefore, the absolute minimum of F can occur at n = 4 although it does depend on 1]@]]/[]@,]1. The situation gets more complicated if the cubic term is absent because Fl(n) always takes its minimum at n = 5 [due to the term Z(4)(n)] . Therefore, if ]I$,113 ]]q5]],then Fl(n) governs the behavior of F, which allows the absolute minimum of the potential at n = 4. On the other hand, for the case of (iii) (n = 2 and 4) the function F is given by
F= ~h2114211 - &,LV>IIdI II@,11 +FI@). The relevant symmetry breaking pattern via SO(4) X SO(6) (n = 4) IS realized even in the absence of the cubic term h, tr(@) because A(5) < A(4) < A(2) and then the term proportional to A(n) plays the same role as that of the cubic term Cc3)(n) [because h, > 0, h, > 0; see (6)] . The explicit conditions are 534) < F(2), Z(5)
andDl+P2>B1
-P,>O
or&
-P2
leading to p2 > 0 *6 . The inclusion of the cubic term makes the vacuum more stable if X4 and h, are appropriately adjusted so as to produce the absolute minimum at n = 4. Finally, notice that for the breaking pattern via SO(5) X SO(5), the vev of @is given by diag (@ = (u, -u, U, -u, U, -u, U, -u, U, -u) in the basis of (Sa) and (8b). *6 The condition of p2 > 0 stated in footnote 3 is actually for the case (i) and (ii). This case (iii) automatically ment in footnote 3 because h2 > 0 if we assume II@11 > It@sll, which is related to footnote 4.
References [l]
[2]
[3] [4] [5] [6] [7] [8]
38
H. Fritzsch and P. Minkowski, Ann. Phys. 93 (1975) 193; M.S. Chanowitz, J. Ellis and M.K. Gaillard, Nucl. Phys. B128 (1977) 506; H. Georgi and D.V. Nanopoulos, Nucl. Phys. B155 (1979) 52. P. Ramond, Caltech preprint CALT-68-709 (February 1979); R. Barbieri et al., Phys. Lett. YOB (19 80) 9 1; E. Witten, Phys. Lett. YlB (1980) 81; M. Magg and Ch. Wetterich, Phys. Lett. 94B (1980) 61. Q. Shafi and Ch. Wetterich, Phys. Lett. 85B (1979) 52; Q. Shafi, M. Sondermann and Ch. Wetterich, Phys. Lett. 92B (1980). M. Yasub, Phys. Lett. YlB (1980) 85. F. Buccella, H. Ruegg and CA. Savoy, Phys. Lett. 94B (1980) 49 1; M. Yasue, INS-preprint INS-rep. -387 (August 1980); to be published in Phys. Rev. D (1981). M. Yasue, in preparation; G. Lazarides, M. Magg and Q. Shafi, Phys. Lett. 97B (1980) 87. M. Yasue, Prog. Theor. Phys. 65 (1981) 708; G. Lazarides, Q. Shafi and C. Wetterich, Nucl. Phys. B181 287. G. Lazarides, M. Magg and Q. Shafi, Phys. Lett. 97B (1980) 87; F. Buccella, H. Ruegg and C.A. Savoy, Nucl. Phys. B169 (1980) 68.
ensures
the state-